Feedback Linearization for Input-saturation Nonlinear

Indian Journal of Science and Technology, Vol 8(S1),97-102, January 2015

ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 10.17485/ijst/2015/v8iS1/57703

Feedback Linearization for Input-saturation Nonlinear System Based on T-S Fuzzy Model Wang Fa-guang(王法广)1, Wang Hong-mei(王洪梅)1*, Park Seung-kyu2, Wang Xue-song(王雪松)1, Sanghyuk Lee3,4 1

School of information and electrical engineering, China university of mining and technology, Xuzhou 221116, China; [email protected] 2 School of Mechatronics, Changwon National University, Changwon 641-773, Korea 3 Department of Electrical and Electronic Engineering, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China 4 Centre for Smart Grid and Information Convergence, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China

Abstract Considering input saturation problem of nonlinear system, a linearized model of multi-inputs nonlinear system is proposed in this paper. The final linear model has prescribed poles and has the same convergence nearby the designed equilibrium points. After this, the linear control theorem can be applied. During the calculation of linearization, T-S (Takagi Sugeno) fuzzy model and pole placement method were utilized. Pole placement just was applied only once for the final model comparing the traditional case where it was designed for every fuzzy rule. This means fewer LMIs (linear matrix inequality) will be needed and its solution will be guaranteed as much as possible. In this paper, nonlinear system will be transferred to T-S fuzzy model first. Note that the T-S fuzzy model is still nonlinear. Then, by employing a series of transfer matrix, nonlinear T-S fuzzy model will be transferred into a nearly linear form accompanied with only one nonlinear part. Finally, by designing a proper controller, linear pole placement method is used and the designed linearization controller gains can be calculated out with LMIs.

Keywords: T-S Fuzzy Control, Linearization, LMIs, Pole Placement, Saturation Nonlinear System

1. Introduction Takagi-Sugeno (T-S) fuzzy theorem is proposed by Takagi and Sugeno in 19851. In the equilibrium points, measurable variables compose fuzzy rules and the corresponding membership functions. he function of system states and inputs is applied as consequent condition of if-then fuzzy rules which usully is linear models. T-S fuzzy local model is formed by the product of consequent model and its corresponding membership function. he overall

* Author for correspondence

T-S model is the summation of its local models. By the application of PDC (parallel distributed compensation), controllers of every local model can be combined with its corresponding membership function as the total nonlinear system controller. By this way, T-S fuzzy control is an eictive way for nonlinear system controller design. Meanwhile, many nonlinear dynamic systems can be represented by Takagi-Sugeno fuzzy models. In fact, it is proved that Takagi-Sugeno fuzzy models are universal approximators. his T-S fuzzy model is not only used

Design and Implementation of Smart Farm System Based on Hybrid App

for fuzzy controller but also used for modeling dynamic model of control object. PDC is one of the useful control methods for T-S fuzzy model2,3, especially, ater LMI (Linear Matrix Inequality) can be calculated in computer. he history of the so-called parallel distributed compensation began with a model-based design procedure proposed by Kang and Sugeno e.g.4. he PDC ofers a procedure to design a fuzzy controller from a given T-S fuzzy model5. he controller parameters can be directly calculated by LMI. In this work, linear pole placement method has been utilized for the nonlinear sytem with input saturation. In the control of nonlinear systems using linear control theories, the feedback linearization technique is popular. he feedback linearization usually involves a state coordinate change and feedback. Feedback linearization via static feedback has been thoroughly studied 6,7. Feedback linearization of discrete-time nonlinear system is more diicult since the method is originally based on the partial diferential operators like Lie brackets and Lie derivatives. More recently, the use of dynamic feedback has been investigated, in the hope of augmenting the class of linearizable systems8,9. Since these studies are based on the profound mathematical background, they are not intuitive at all and diicult to know the exact boundary between linearizable and nonlinearizable systems. So that, inding veriiable necessary and suicient conditions to characterize the class of such linearizable systems is still an open problem. In hardware system, there are many constraints because hardware limitations. Some systems have to change the control mode from one to another and so on. For these reasons, the input of control object and the output of controller may unequal. his can result in the response of closed-loop system worse, and it is called windup. If ignored this kind of nonlinearity problem during system design, the system overshoot cannot be suppressed well and the system may be unstable. here are many studies in this area10. As windup usually caused by integral, Buckley raised an anti-reset windup method. he error of controller output and object output is used as feedback to complement the system11. Condition technique is shown in the work of Hanus and his partners irst. his method re-computes reference inputs which keep controller output out of saturation area meanwhile tracking the new reference input. So the control object input will be same with controller output, and windup is eliminated12. Still, there are anti-windup controller based on observer13,

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internal model control14, saturation feedback control15, dynamic complement16 and so on. Variable structure control is proposed to prevent integrator windup by preset parameter17. Variable structure anti-windup controller has a good perfermance, however the preset paremeter is based on experimance. Stability and robust are diicult to discuss. In this paper, based on T-S fuzzy theorem, inputsaturation nonlinear system is transformed to a linear system which can assign the poles in a desired disk. his linear system will keep the same convergence with the nonlinear system nearby the equillibrium points. Ater this procedure, linear control theorem can be applied for controller design. Proposed method has meanings in stability and robust analysis. Next part will introduce the handling of nonlinear system by T-S fuzzy theorem. he third part is about the solution of input-saturation problem for nonlinear system and the inal results will be shown. he fourth and ith parts are simulation example and conclusion.

2. T-S Fuzzy Transfermation For Multi-Inputs Nonlinear System Traditional linear pole placement cannot be applied to nonlinear system directly. In this paper, T-S fuzzy acted as mediator between linear pole placement and nonlinear system. he i-th IF-THEN rule of the T-S fuzzy model for a nonlinear system is Rule i : If w1 (t ) is Fi1 and ⋯ and wg (t ) is Fig , then xɺ (t ) = Ai x(t ) + Bi u (t ) ,

(1)

n× n n× m where, i = 1, 2,..., L , Ai ∈ R , Bi ∈ R , L is the number of rules, and w1 (t ), w2 (t ),..., wg (t ) are premise variables.

he overall fuzzy system is L

∑ µ (w(t )) { A x(t ) + B u (t )} i

xɺ (t ) =

i

i

i =1

L

∑ µ (w(t )) i

i =1

where,

w(t ) =  w1 (t ), w2 (t ),..., wg (t ) 

(2) . Let

Indian Journal of Science and Technology

Won-IL Seo, Seung-Mi Moon, Sook-Youn Kwon, Jin-Hee Park and Jae-Hyun Lim

hi ( w(t )) =

µi ( w(t ))

L

Zɺ (t ) = ∑ hi ( w(t )) { Aci Z (t ) + Bci u (t )}

L

∑ µ (w(t ))

i =1

i

i =1

L

xɺ (t ) = ∑ hi ( w(t )) { Ai x(t ) + Bi u (t )} i =1

. Membership function hi ( w(t )) should satisfy:

(3)

Now problem is to derive Ti and u(t) which make the T-S fuzzy model (9) linear. Under the assumption 1, Ti is obtained by following steps: Structure a matrix M as M i = [bi1

L

i =1

(4)

he following assumption is required for the inal model calculation in this paper:

2.1 Assumption 1 If all T-S fuzzy n-dimension linear models are controllable, the following condition should be satisied:

⋯ Ai bi 2 ⋯]

... A Bi  = n .

(5)

To deal with the coordinate change, the i-th rule of the fuzzy model for the nonlinear system is Rule i: If w1 (t ) is Fi1 and ⋯ and wg (t ) is Fig , then xɺ (t ) = Ai x(t ) + Bi u (t ) , for i = 1, 2,..., L and z (t ) = Ti x(t )

(6)

(10)

where, bij is the elements of Bi = [bi1 bi 2 ... bim ] in (1). For searching M i , start from bi1 and end at rank(M)=n. If Aµi +1bik makes vectors before this vector and this vector relative, then delete this vector and start with bi (k +1) . he inverse of M is M i−1 = eiT11 eiT21

n −1 i

Ai bi 2

µ2

Ab

i

Ai2 bi1 ⋯ Aiµ1bi1 bi 2

Ai bi1

2 i i1

∑ h (w(t )) = 1.

Ai Bi

(9)

, nonlinear system can be described

as

rank  Bi

.

eiT12 ⋯ eiT1µ1 T

eiT22 ⋯ eiT2 µ 2 ⋯ .

(11)

For the multiple-input case, Ti is obtained as follows: Ti = [(ei1µ1 )T

(ei1µ1 Ai )T

(ei 2 µ 2 )T

(ei 2 µ 2 Ai )T

⋯ (ei1µ1 Aiµ1−1 )T ⋯ (ei 2 µ 2 Aiµ 2 −1 )T ⋯]

(12)

hen the above state transformation Ti changed the i-th linear system of fuzzy system into the controllable canonical form

for i = 1, 2,..., L .

Zɺ (t ) = Aci Z (t ) + Bci u (t )

By substitute state x(t) of (6) into (1), the linear model of i-th rule can be Zɺ (t ) = Aci Z (t ) + Bci u (t )

(7)

−1 Bci = Ti Bi . he overall fuzzy where, Aci = Ti AT i i , coordinate changed state is L

Z (t ) = ∑ hi ( w(t ))Ti x(t ) i =1

(8)

which is considered as the summation of the states x(t) through transformation Ti with the proportion of membership hi . he overall fuzzy system is

Vol 8 (S1) : January 2015 : www.indjst.org

(13)

where,  0  0   ⋮   0  ain 1  l Aci =          

1

0

0 ⋮ 0

1 ⋮ 0

ainl 2

ainl 3

⋯ ⋯ ⋱

0

0 0 0 ... 0

0 ⋮ 1

0 0 0 ... 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ... 0

...

⋯ ⋯ ainl nq

⋮

× × × ... × ⋱

⋮

0 0 0 ... 0

0

1

0

0 0 0 ... 0

0

0

1

⋮ 0

⋮ 0

⋮ 0

ain ( nk +1)

ain ( nk + 2)

⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ... 0 × × × ... ×

...

⋯ ⋯ ⋱

⋯ ain ( nk + 3) ⋯

          0  0   ⋮  1   ainn  

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Design and Implementation of Smart Farm System Based on Hybrid App

Zɺ (t ) = Acn Z (t ) + Bcn v(t )

0 ... 0   0  ⋮ ⋮ ⋱ ⋮    0 0 ... 0    bcinl 1 bcinl 2 ... bcinl m  , a Bci =  inl nq presents the 0 0 ... 0     ⋮ ⋮ ⋱ ⋮    0 ... 0   0  bcin1 bcin 2 ... bcinm 

where,

nq -th element of the nl -th row in the i-th rule and × presents the other values. his mark is same to the elements of Bci .

he overall fuzzy system (9) can be reform as L

Zɺ (t ) = ∑ hi ( w(t )) { Aci Z (t ) + Bci u (t )} i =1

z2 (t )    0      ( ) z t 3 0        ⋮  ⋮     ( ) z t ( nl −1)    0   L n   L m   ∑∑ hi ain j z j (t )   hi bcinl k uk (t )  l ∑∑  i =1 j =1   i =1 k =1      ⋮ = + ⋮       z( nk + 2) (t ) 0      0 z( nk + 3) (t )         ⋮ ⋮        0 zn (t )     L m  L n    hi bcink uk (t )  ∑∑ h a z t ( )  ∑∑ i inj j   i =1 k =1   i =1 j =1 

(14)

L

2.1.1 heorem 1 Under the assumption 1, if m

∑∑ h b i =1 k =1

i cinl k

L

n

uk (t ) = −∑∑ hi ainl j z j (t ) i =1 j =1

(15)

for each nl has the solution for uk (t ) , then (9) can be the form of

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0 0  ⋮  0 0  Acn =  0  0 ⋮  0  0

0 ⋮  0  × Bcn =  0  ⋮ 0  ×

1 0 ⋮ 0 0 0 0 ⋮ 0 0

0 1 ⋮ 0 0 ⋮ 0 0 ⋮ 0 0

⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯

0 0 ⋮ ⋯ 1 0 ⋱ 0 0 ⋮ ⋯ 0 0

0 0 ⋮ 0 0

0 0 ⋮ 0 0

0 0 ⋮ 0 0

1 0 ⋮ 0 0

0 0 ⋮ 0 0 ⋮ 0 1 ⋮ 0 0

⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯

0 0  ⋮  0 0   0  0 ⋮  1 0 

0 ... 0  ⋮ ⋱ ⋮  0 ... 0   × ... × 0 ... 0   ⋮ ⋱ ⋮ 0 ... 0   × ... ×

, Z(t) is an n-dimension vector, × replace 1 or some values and anti-windup controller is included in v(t) which is a m-dimension vector.

3. The Linear Model of InputSaturation Nonlinear System

hi ( w(t )) = 1 where, uk (t ) is the k-th input and remind that ∑ i =1 from (4). he result can be represented in the following theorem:

L

(16)

In many equipments, there is inputs saturation problem. heorem 1 can’t work well under input saturation case because it is a feedback controller design method. Considering of this, an anti-windup controller is needed. Consider the saturation input u (t ) ≥ ulim  ulim ,  v(t ) = sat (u (t )) =  u (t ) −ulim < u (t ) < ulim −u , u (t ) ≤ −ulim  lim

(17)

u(t) is input before saturation segment and ulim > 0 is input limitation. Degree of feedback saturation is deined as:

Indian Journal of Science and Technology

Won-IL Seo, Seung-Mi Moon, Sook-Youn Kwon, Jin-Hee Park and Jae-Hyun Lim

 ulim KZ (t ) > ulim  KZ (t ) ,  −ulim < KZ (t ) < ulim S (t ) =  1,  u − lim , KZ (t ) < −ulim  KZ (t )

Table 1. IPMSM parameters Pole pair number P 2

(18)

Deine

d-axis inductance Ld

42.44 mH

q-axis inductance Lq

79.57 mH

Stator resistance R

1.93 Ω

Motor ineria

d 0 = min{S (t ), ∀Z ∈ D}

(19)

where, D is a compact space and let a matrix S0 is consisted by element d0 or 1. Set v(t)=KZ(t),

(20)

3.1 Corollary 1 System (16) is D-stable if and only if there exists a symmetric matrix X>0 such that − rX  qX + X ( A + B S K ) T cn cn 0 

qX + ( Acn + Bcn S 0 K ) X   < 0. − rX 

where, q is the pole placement disk center and r is radius. he proof is clearly based on the work of20. Equation (21) can be solved by LMI, then v(t) in (20) can be designed with anti-windup ability. he inal result can be gived as:

Under the assumption 1, nonlinear system can be linearized by the controller of (15). If there is input saturation of (17), (21) can be used to design the v(t) of (16). he inal controller is m

k =1

(22)

which can be applied to nonlinear system to guarantee it stable.

4. Numerical Example Based on the mathematic model of an IPMSM in the d-q synchronously rotating reference frame mentioned in21. Parameters are shown in Table 1.

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0.003 kgm 0.001 Nm/rad/sec

Bm

Magneic lux constant ø f

0.311 volts/rad/sec

Set x1 (t ) = iq (t ) , x2 (t ) = id (t ) , x3 (t ) = w(t ) , u1 (t ) = uq (t ) , u2 (t ) = ud (t ) , e1 (t ) = x1 (t ) − x1r (t ) , e2 (t ) = x2 (t ) − x2r (t ) u1e (t ) = u1 (t ) − u10 (t ) , and e3 (t ) = x3 (t ) − x3r (t ) , , u2e (t ) = u2 (t ) − u20 (t ) . Here reference values are x1r = 0.0209 , x2 r = 1.5817 and x 3r =15.8144 . he error dynamic system is derived out as

+ eɺ2 (t ) = + eɺ3 (t ) =

3.1.1 heorem 2

u (t ) = ∑ uk (t ) + v(t )

Fricion coeicient

eɺ1 (t ) = −

(21)

2

Jm

+

Pψ f + PLd x2 (t ) PL R e1 (t ) − d x3r e2 (t ) − e3 (t ) Lq Lq Lq

1 u10 (t ) Lq PLq Ld

x3r e1 (t ) −

PLq x1 (t ) R e2 (t ) + e1 (t ) Ld Ld

1 u20 (t ) Ld 3P(ψ f + ( Ld − Lq ) x2 r (t )) 2Jm 3P ( Ld − Lq ) x1 (t ) 2Jm

e2 (t ) −

e1 (t ) Bm e3 (t ) Jm

where, x1 (t ) and x2 (t ) are selected as premise variables. he input saturation degree is 0.8. here are 4 fuzzy rules for this system. he coordinate change matrixes Ti can be get from (12), and linearization controller can be calculated from (15). Set the poles in a disk with origin (-2000,0) and radius r=1000. he anti-windup controller gains of (20) is K=[-362.1858,-40.4408,0;0,0,-22.0059]. Nominal system eigenvalues are -13.3883, -27.0525 and -22.0059. A set of random change values from -0.5 to 0.5 is used as disturbance which is added in state x1 (t ) before the integral. Two saturation inputs are both of form (17) where ulim = 15 for the error system. he inal controller form is (22). he simulation results of e1 (t ) compared with traditional Lyapunov PDC T-S fuzzy control result

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101

Design and Implementation of Smart Farm System Based on Hybrid App

is in Figure 1. he other satae-results are absent here for economizing space. System output errors are almost zero. For the existance of input saturation limitation, settling time is about 0.2s.

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6. Su R. On the linear equivalents of nonlinear systems. Syst Contr Lett. 1982; 2(1):48–52. 7. Isidori A. Nonlinear control systems. 3rd ed. London: Springer-Verlag Ltd; 1995. 8. Li Y-F, Liu X-D, Xing G.-Q Discrete-time LQ optimal control of satellite formations in elliptical orbits based on feedback linearization. Acta Astron. 2013; 83(1):125–31.

9. Aboushady A, Ahmed K, Finney S. Linearized large signal modeling, analysis, and control design of phasecontrolled series-parallel resonant converters using state feedback. IEEE Trans Power Electron. 2013; 28(8):3896– 911.

Figure 1. he performance of e1 (t ) .

5. Conclusions Based on T-S fuzzy and pole placement, an anti-windup controller is proposed for saturation nonlinear multiinputs system in this paper and the controller parameters can be calculated by LMIs. he proposed controller was applied for a PMSM model as a test. he comparing results showed that the proposed controller of this paper gave a good control performance.

6. Acknowledgement his research was inancially supported by “the Fundamental Research Funds for the Central Universities” (Grant No.2013QNB26).

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10. Sajjadi K, Jabbari F. Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup. Automatica 2013; 49(3):762–9. 11. Buckley P. Designing override and feedforward controls. Cont Engin. 1971; 18(8): 48–51. 12. Hanus R, Kinnaert M, Luyben W. Conditioning technique, a general anti-windup and bumpless transfer method. Automatica.1987; 23(6):729–39.

13. Navneet K, Prodromos D. Observer-based anti-windup scheme for non-linear systems with input constraints. Int J Contr. 1999; 72(1):18–29. 14. Zheng A, Kothare M, Morari M. Anti-windup design for internal model control. Int J Contr. 1994; 60(5):1015–24. 15. Levine W. he control handbook. FL, USA: CRC Press; 1996.

16. Zaccarian L, Li Y-P, Weyer E., Cantoni M, Teel A. Antiwindup for marginally stable plants and its application to open water channel control systems. Contr Eng Pract. 2007; 15(2):261–72. 17. Herrmann G, Menon P, Turner M. Anti-windup synthesis for nonlinear dynamic inversion control schemes. Internat of Rob and Nonlin Cont. 2010; 20(13):1465–82. 18. Wang F-G, Park S, Yoon T, Kwak G. Linearization of discrete-time T-S fuzzy model with multiple input. Proceedings of the International Conference on Intelligent Systems Design and Applications; 2010 Nov 29 – Dec 1 ; Cairo. Egypt. 19. Yoon T, Wang F-G, Park S, Kwak G, Ahn H. Linearization of T-S fuzzy systems and robust H-ininity control. J of Centr South Univer of Technol. 2011; 18(1):140–5.

20. Chilali M, Gahinet P. Design with pole placement constraints: An LMI approach. IEEE Trans Automat Contr. 1996; 41(3):358–67. 21. Nasir U, Azizur R. High-Speed control of IPMSM drives using improved fuzzy logic algorithms. IEEE Trans Ind Electron. 2007; 54(1):190–9.

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